Suppose we have a controllable discrete time linear system \begin{align*} x_{t+1} = Ax_t + Bu_{t} \end{align*} In order to design a stabilizing LQR controller with respect to supply rate $l(x,u)=x^{\top}Qx + u^{\top}Ru$ with positive definite $Q$ and $R$, we can solve the discrete time Algebraic Riccati Equation (DARE) \begin{align*} \begin{bmatrix} I \\ K \end{bmatrix}^{\top} \begin{bmatrix}A^{\top}PA+Q-P & A^{\top} P B \\ * & B^{\top} P B + R \end{bmatrix} \begin{bmatrix} I \\ K \end{bmatrix} = 0 \end{align*} for $K$ and $P$. My question is regarding the uniqueness of $K$ given a $P$. In other words, let $\bar{K}$ and $\bar{P}$ be the solutions of the equation above. Then, does it hold that for any $K \neq \bar{K}$, \begin{align*} \begin{bmatrix} I \\ K \end{bmatrix}^{\top} \begin{bmatrix}A^{\top}\bar{P}A+Q-\bar{P} & A^{\top} \bar{P} B \\ * & B^{\top} \bar{P} B + R \end{bmatrix} \begin{bmatrix} I \\ K \end{bmatrix} \succ 0. \end{align*} Random simulations do seem to indicate this is true.
Thank you!
